Database
REAL AND COMPLEX NUMBERS
Elementary integer functions
Integer powers
ltexp1d
Metamath Proof Explorer
Description: Elevating to a positive power does not affect inequalities. Similar to
ltmul1d for exponentiation of positive reals. (Contributed by Steven
Nguyen , 22-Aug-2023)
Ref
Expression
Hypotheses
ltexp1d.1
⊢ φ → A ∈ ℝ +
ltexp1d.2
⊢ φ → B ∈ ℝ +
ltexp1d.3
⊢ φ → N ∈ ℕ
Assertion
ltexp1d
⊢ φ → A < B ↔ A N < B N
Proof
Step
Hyp
Ref
Expression
1
ltexp1d.1
⊢ φ → A ∈ ℝ +
2
ltexp1d.2
⊢ φ → B ∈ ℝ +
3
ltexp1d.3
⊢ φ → N ∈ ℕ
4
rpexpmord
⊢ N ∈ ℕ ∧ A ∈ ℝ + ∧ B ∈ ℝ + → A < B ↔ A N < B N
5
3 1 2 4
syl3anc
⊢ φ → A < B ↔ A N < B N